A Pade approximation is an approximation of a function represented by a rational polynomial. Pade approximations can have any order numerator paired with any order denominator. Any function has an infinite number of pade approximations. Pade approximations are used for representing time delays in models. This is important because neglecting to model the time delays of physical systems can lead to closed-loop oscillations when incorporated into a controller. The frequency delay equation can be represented by a Pade polynomial to allow systems with a time delay to be used for methods such as LQR, H-\(\infty\), and Root-Locus. It captures the time delay dynamics with a transfer function that has a finite number of states. The equation for a Pade approximation with numerator order \(m\) and denominator order \(n\) is
$$R(x)=\frac{\sum^m_{j=0}a_jx^j}{1+\sum^n_{k=0}b_kx^k}$$
This rational polynomial is then equated to the Taylor series approximation of your desired function and then solved for the coefficients of \(a\) and \(b\), only considering terms up to \(n+m\). Equal order Pade approximations (\(m=n\)) are the only approximations that are an all-pass filter, as in they affect phase alone. All the other approximations affect the system gain therefore are not desirable for representing a time delay. When using a Pade approximation you need to be sure that the phase of the approximation matches the phase of the system up to the desired cut-off frequency. The N/M pade approximation agrees with the first M+N terms in the taylor series.
[[Pade Approximations of e^x]]
Exponential Matrix Technique – uses a Pade(3,3) approximation
[[Disadvantages of the Taylor Series]] – Pade approximations follow the function for longer
[[Calculating the Pade Approximation of sin(x)]]
[[RASCAL Baseline Control System]]- used pade approximations for sensor and computational delays
[[MCLAWS Simulink Model]] – used Pade approximations for sensor phase lag